FOM: Cantor'sTheorem & Paradoxes & Continuum Hypothesis

Kanovei kanovei at
Mon Feb 12 14:20:45 EST 2001

>From: Neil Tennant <neilt at>
>Cantor's theorem does not depend on the assumption that
the power set of the continuum exists.

Once again, the proof has two important issues, 
1) technically - the diagonal method (which in fact was used, 
perhaps, earlier by Du Bois Reymond in "scaling" sequences, 
near 1872)
2) foundationally - the postulate that all elements of P(N) 
(or P(X),for "arbitrary" X - which was not much meaningful 
for a XIX century mathematician) 
are "already given" and no new ones can appear in the course 
of the proof. 

The rest is a couple of lines of ordinary transformations. 

Something similar to 2) appears in many paradoxes, say those 
of Liar and Barber in the form of a mess between "has been" 
and "has been and will always be".


More information about the FOM mailing list